Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

Let's say I sum an exponential function $e^{\imath \left(k-k^{\prime}\right) x_{i}}$ over a chain system where every member of the chain is of the same type, e.g.,

A-A-A-...-A-A (total of N sites)

I would obtain (typical textbook example)

$$\sum_{i=1}^{N} e^{\imath \left(k-k^{\prime}\right) x_{i}} = \frac{1}{N} \delta _{k;k^{\prime}}$$

When we consider alternating site types and sum only over sites of type A, i.e.,

A-B-A-B-...-A-B

what would the sum look like?

$$\sum_{i\in A} e^{\imath \left(k-k^{\prime}\right) x_{i}} = ?$$

Note: The sum is now over $N_{A}=\frac{N}{2}$ sites. Also, the fact that it's a chain is just an example, I could also consider 2D or 3D grids like in crystals. The only difference would be that $k$, $k^{\prime}$ and $x_{i}$ are vectors then.

share|improve this question
    
Hint: how do you sum geometrical series? –  Piotr Migdal Jul 11 '12 at 15:04
    
I realized after trying to figure it out that I asked the wrong question... –  Robert Jul 12 '12 at 18:01
add comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.